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    <article id="post-高等数学/极限与连续/函数的性质" class="article article-type-post" itemscope itemprop="blogPost">
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    <a href="/2019/06/23/高等数学/极限与连续/函数的性质/" class="article-date">
  <time datetime="2019-06-22T21:43:13.000Z" itemprop="datePublished">2019-06-23</time>
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    <a class="article-category-link" href="/categories/高等数学/">高等数学</a>►<a class="article-category-link" href="/categories/高等数学/极限与连续/">极限与连续</a>
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      <a class="article-title" href="/2019/06/23/高等数学/极限与连续/函数的性质/">函数的性质</a>
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        <h2 id="唯一性"><a href="#唯一性" class="headerlink" title="唯一性"></a>唯一性</h2><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">如果数列收敛,那么他的极限唯一</span><br></pre></td></tr></table></figure>
<script type="math/tex; mode=display">
若 \lim_{n\rightarrow\infty}a_n = A</script><script type="math/tex; mode=display">
\lim_{n\rightarrow\infty}a_n = B</script><p>则 A = B</p>
<h2 id="保号性"><a href="#保号性" class="headerlink" title="保号性"></a>保号性</h2><script type="math/tex; mode=display">
设 \lim_{x \to a}f(x) = A 
 \begin{cases} > 0 \\\\ < 0  \end{cases}</script><script type="math/tex; mode=display">
则 \exists \delta > 0, 当 0 < |x -a | < \delta时 \\
f(x) \begin{cases}  > 0 \\\\ < 0  \end{cases}</script><p>例 1 </p>
<script type="math/tex; mode=display">
f^{'}(1) = 0, \lim_{x \to 1}{\frac{f^{'}(x)}{(x-1)^3}} = -2, x = 1 是什么点?</script><script type="math/tex; mode=display">
因为 原式 = -2 < 0 \\\\   
所以 \exists \delta > 0, 当 0 < |x -1| < \delta时 \\\\
原式 < 0 \\\\
可知 \begin{cases}  f^{'}(x) > 0 , x \in (1 - \delta , 1) \\\\ 
 f^{'}(x) < 0 , x \in (1, 1+\delta) \end{cases} \\\\
 故: x=1 时是 极大点</script><h2 id="存在性"><a href="#存在性" class="headerlink" title="存在性"></a>存在性</h2><h3 id="数列型"><a href="#数列型" class="headerlink" title="数列型"></a>数列型</h3><script type="math/tex; mode=display">
if a_n <= b_n < c_n; \lim a_n = \lim c_n = A \\\\  
则 \lim b_n = A   (夹逼定理)</script><h3 id="函数型"><a href="#函数型" class="headerlink" title="函数型"></a>函数型</h3><script type="math/tex; mode=display">
if f(x) <= g(x) <= h(x); \lim_{x \to a}f(x) = \lim_{x \to a}h(x) = A \\\\ 
则 \lim_{x \to a }g_n = A</script><script type="math/tex; mode=display">
证:设 \lim_{x \to a}f(x) = \lim_{x \to a}g(x) = A \\\\  
故\forall \varepsilon > 0, \exists\delta_1 > 0, 满足 0<|x-a|<\delta_1时\\\\  
|f(x) - A| < \varepsilon \\\\  
A-\varepsilon < f(x) < A+\varepsilon  \quad(1)\\\\  
同理可知 存在 \exists\delta_2 > 0, 满足 A-\varepsilon < h(x) < A+\varepsilon \quad(2) \\\\  
取 \delta = min\{ \delta_1, \delta_2\}, 则 
满足 0 < |x - a | < \delta 时, \\\\
A+\varepsilon < f(x) <= g(x) <= h(x) < A+\varepsilon \\\\ 
即| g(x) - A | < \varepsilon, \\\\ 
所以 \forall\varepsilon > 0,当0<|g(x)  - A |<\delta 时 \\\\
\lim_{x \to a}f(x)　＝　Ａ</script><h2 id="有界性"><a href="#有界性" class="headerlink" title="有界性"></a>有界性</h2><script type="math/tex; mode=display">
\lim_{n\rightarrow\infty}a_n = A, 则\exists M > 0 ,满足 | a_n| < M</script><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">如果数列收敛,那么数列一定有界</span><br></pre></td></tr></table></figure>

      
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    <article id="post-数据结构与算法设计/小甲鱼/抽象数据类型" class="article article-type-post" itemscope itemprop="blogPost">
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        <!-- Table of Contents -->
        
        <pre class="mermaid">graph LR;
A[ADT] --> B(定义)
B --> C(数据模型)
B-->D(支持的运算)
A -->E(实现)
E-->G(逻辑实现)
E-->H(实现方式)
A-->F(应用)</pre>




      
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    <article id="post-数据结构与算法设计/小甲鱼/表" class="article article-type-post" itemscope itemprop="blogPost">
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    <a class="article-category-link" href="/categories/数据结构与算法/">数据结构与算法</a>►<a class="article-category-link" href="/categories/数据结构与算法/小甲鱼/">小甲鱼</a>
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        <p>ADT表</p>
<h2 id="定义"><a href="#定义" class="headerlink" title="定义"></a>定义</h2><pre class="mermaid">graph LR;
A[表] --> B(定义)
B --> C(数据模型)
C-->M(同一种类型的元素的有限序列)
B-->D(支持的运算)
D-->N(长度Length)
D-->O(是否空)
D-->P(元素的位置)
D-->Q(前驱/后继)
A -->E(实现)
E-->G(逻辑实现)
E-->H(实现方式)
H-->I(数组实现)
H-->J(指针实现)
H-->K(间接寻址实现)
H-->L(游标实现)
A-->F(应用)</pre>


      
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    <article id="post-高等数学/极限与连续/无穷小和无穷大" class="article article-type-post" itemscope itemprop="blogPost">
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    <a href="/2019/06/12/高等数学/极限与连续/无穷小和无穷大/" class="article-date">
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    <a class="article-category-link" href="/categories/高等数学/">高等数学</a>►<a class="article-category-link" href="/categories/高等数学/极限与连续/">极限与连续</a>
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        <h2 id="无穷小"><a href="#无穷小" class="headerlink" title="无穷小"></a>无穷小</h2><script type="math/tex; mode=display">
当\lim_{x\rightarrow a}f(x) = 0, 称函数 f(x) 为当x \to a 时的无穷小</script><script type="math/tex; mode=display">
0是无穷小,但无穷小不一定为0 \\\\  
f(x) \neq 0 , f(x) 是否无穷小与x的趋向有关</script><h2 id="无穷小相比"><a href="#无穷小相比" class="headerlink" title="无穷小相比"></a>无穷小相比</h2><script type="math/tex; mode=display">
设 \alpha 为无穷小, \beta 为无穷小</script><script type="math/tex; mode=display">
\lim_{}\frac{\alpha}{\beta} = 0 , 则称\beta为\alpha 的高阶无穷小</script><script type="math/tex; mode=display">
\lim_{}\frac{\alpha}{\beta} = k(常数 \neq  0) , 则称\beta为\alpha 的同阶无穷小</script><script type="math/tex; mode=display">
特例: \lim_{}\frac{\alpha}{\beta} = 1 , 则称\beta为\alpha 的等价无穷小 ,  \alpha \sim \beta</script>
      
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    <article id="post-高等数学/极限与连续/函数的极限" class="article article-type-post" itemscope itemprop="blogPost">
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    <a href="/2019/06/11/高等数学/极限与连续/函数的极限/" class="article-date">
  <time datetime="2019-06-11T00:24:28.000Z" itemprop="datePublished">2019-06-11</time>
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    <a class="article-category-link" href="/categories/高等数学/">高等数学</a>►<a class="article-category-link" href="/categories/高等数学/极限与连续/">极限与连续</a>
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        <h2 id="定义"><a href="#定义" class="headerlink" title="定义"></a>定义</h2><h3 id="自变量趋向于有限值"><a href="#自变量趋向于有限值" class="headerlink" title="自变量趋向于有限值"></a>自变量趋向于有限值</h3><h3 id="定义-1"><a href="#定义-1" class="headerlink" title="定义"></a>定义</h3><p>若y = f(x) 在x =a 的去心邻域内有定义, (总能找到$\delta$)</p>
<script type="math/tex; mode=display">
当 \forall\large\varepsilon > 0, \exists\delta > 0, 当 0 <|x-a| < \delta时</script><script type="math/tex; mode=display">
|f(x) - A| < \varepsilon</script><p>则称</p>
<script type="math/tex; mode=display">
\lim_{x\rightarrow a}f(x) = A</script><h3 id="要点"><a href="#要点" class="headerlink" title="要点"></a>要点</h3><script type="math/tex; mode=display">
 x \to a  \implies  x  \neq  a</script><script type="math/tex; mode=display">
\lim_{x\rightarrow a} f(x) 与 f(a) 无关</script><script type="math/tex; mode=display">
x \to a \implies \begin{cases} x \to a^- \\\\\ x \to a^+  \end{cases}</script><script type="math/tex; mode=display">
a的去心邻域, 邻域半径 \delta >  0,   \forall\varepsilon > 0, \exists\delta > 0, 当x \in (a - \delta, a) 时,</script><script type="math/tex; mode=display">
充要条件 \lim_{x\rightarrow a}f(x) \exists \Longleftrightarrow  f(a^-) , f(a^+) \exists 且相等</script><h3 id="例题"><a href="#例题" class="headerlink" title="例题"></a>例题</h3><h4 id="例1"><a href="#例1" class="headerlink" title="例1"></a>例1</h4><script type="math/tex; mode=display">
\lim_{x\rightarrow 2}(3x+1) = 7</script><p>证:  </p>
<script type="math/tex; mode=display">
对任意\varepsilon  > 0 \\\\  
若原式成立, 则需满足|f(x) - 7| = |3x+1-7| = |3x-6| = 3|x-2| < \large\varepsilon成立\\\\  
即满足 |x-2| <\frac{ \varepsilon }{3} 成立 \\\\  
取0 < \delta <= \frac{\varepsilon}{3} ,此时满足0 < |x - 2| < \delta <= \varepsilon\\\\  
故等式成立</script><h2 id="自变量趋向于无穷大时函数的极限"><a href="#自变量趋向于无穷大时函数的极限" class="headerlink" title="自变量趋向于无穷大时函数的极限"></a>自变量趋向于无穷大时函数的极限</h2><h3 id="情况-1"><a href="#情况-1" class="headerlink" title="情况 1"></a>情况 1</h3><p>区分正负无穷极限</p>
<script type="math/tex; mode=display">
y = arctanx \\\\  
\lim_{x\rightarrow -\infty} arctanx = -\frac{\pi}{2}</script><script type="math/tex; mode=display">
\lim_{x\rightarrow +\infty} arctanx = \frac{\pi}{2}</script><p><img src="../assets/1.svg" alt=""></p>
<h3 id="情况-2"><a href="#情况-2" class="headerlink" title="情况 2"></a>情况 2</h3><script type="math/tex; mode=display">
y = 2 + e^{-x^2}</script><p><img src="../assets/2.svg" alt="xc"></p>
<h3 id="定义-2"><a href="#定义-2" class="headerlink" title="定义"></a>定义</h3><script type="math/tex; mode=display">
if \quad\forall\varepsilon > 0, \exists X > 0, 当 |x| > X 时  \\\\  
|f(x) - A | < \lim_{x\rightarrow\infty} = \varepsilon,  \\\\</script><script type="math/tex; mode=display">
则 \lim_{x\rightarrow\infty} = A</script><h3 id="例-1"><a href="#例-1" class="headerlink" title="例 1"></a>例 1</h3><script type="math/tex; mode=display">
\lim_{x\rightarrow +\infty} \frac{1}{2x+1} = 0</script><script type="math/tex; mode=display">
证: 对于\forall\varepsilon > 0,若要求 
|f(x) - A | = |\frac{1}{2x+1} - 0| \\\\  
=|\frac{1}{2x+1}| < \varepsilon \\\\  
即 x > \frac{\frac{1}{\varepsilon} - 1}{2} \\\\  
取 X <= \frac{\frac{1}{\varepsilon} - 1}{2} 则满足 \\\\  
x > X, |f(x) - A | < \varepsilon, 从而得证</script>
      
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    <article id="post-高等数学/极限与连续/数列极限的定义" class="article article-type-post" itemscope itemprop="blogPost">
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    <a href="/2019/05/30/高等数学/极限与连续/数列极限的定义/" class="article-date">
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        <h3 id="定义：-Case-varepsilon-—N-语言"><a href="#定义：-Case-varepsilon-—N-语言" class="headerlink" title="定义： Case $\varepsilon$—N 语言"></a>定义： Case $\varepsilon$—N 语言</h3><script type="math/tex; mode=display">
if \forall\large\varepsilon > 0, \exists N > 0 \\\\   
当 n  > N 时， | a_n - A | < \varepsilon</script><script type="math/tex; mode=display">
则 \lim_{n\rightarrow\infty}a_n = A</script><p>例 1 </p>
<script type="math/tex; mode=display">
\lim_{n\rightarrow\infty}\frac{n-1}{2n+1} = \frac{1}{2}</script><p>证: </p>
<script type="math/tex; mode=display">
\forall\large\varepsilon > 0 \\\\  
若要求
\mid \frac{n-1}{2n+1} - \frac{1}{2} \mid = \frac{2(n-1)-(2n+1)}{2(2n+1)} \\\\  
=\mid \frac{-3}{4n+2} \mid \\\\  
< \frac{3}{4n} \\\\  
< \varepsilon  \\\\  
成立 \\\\   
即要求  \frac{3}{4n} < \varepsilon ,\quad n> \frac{3}{4\varepsilon} 成立 \\\\  
取 N = [\frac{3}{4\varepsilon}] \\\\
此时 当 n > N时, 则满足上述要求</script>
      
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        <h2 id="二元情况"><a href="#二元情况" class="headerlink" title="二元情况"></a>二元情况</h2><h3 id="唯一解"><a href="#唯一解" class="headerlink" title="唯一解"></a>唯一解</h3><script type="math/tex; mode=display">
\begin{Bmatrix}
 x_1 -  &   x_2 =  &1 \\\\ 
x_1 + & x_2 =  & 3
\end{Bmatrix}</script><p><img src="../assets/4.svg" alt=""></p>
<h3 id="无解-（直线无交点）"><a href="#无解-（直线无交点）" class="headerlink" title="无解 （直线无交点）"></a>无解 （直线无交点）</h3><script type="math/tex; mode=display">
\begin{Bmatrix}
x_1-  & x_2 = &1  \\\\ 
x_1 - &x_2 =  & 3 
\end{Bmatrix}</script><p><img src="../assets/5.svg" alt=""></p>
<h3 id="无穷解-（两直线重合）"><a href="#无穷解-（两直线重合）" class="headerlink" title="无穷解 （两直线重合）"></a>无穷解 （两直线重合）</h3><script type="math/tex; mode=display">
\begin{Bmatrix}
x_1-  & x_2 = &1  \\\\ 
2x_1 - &2x_2 =  & 2 
\end{Bmatrix}</script><p><img src="../assets/6.svg" alt=""></p>
<h3 id="超定二元方程的近似解"><a href="#超定二元方程的近似解" class="headerlink" title="超定二元方程的近似解"></a>超定二元方程的近似解</h3><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">方程组不相容， 没有精确解</span><br></pre></td></tr></table></figure>
<script type="math/tex; mode=display">
\begin{Bmatrix}
x_1 -  & x_2 = &1  \\\\ 
x_1 +  & x_2 = &3  \\\\ 
x_1 + &2x_2 =  & 5 
\end{Bmatrix}</script><p><img src="../assets/7.svg" alt=""></p>
<h2 id="三元情况"><a href="#三元情况" class="headerlink" title="三元情况"></a>三元情况</h2><h3 id="适定方程组-（唯一解）"><a href="#适定方程组-（唯一解）" class="headerlink" title="适定方程组 （唯一解）"></a>适定方程组 （唯一解）</h3><script type="math/tex; mode=display">
\begin{Bmatrix}
x_1 -  & x_2 = &1  \\\\ 
x_1 +  & x_2 = &3  \\\\ 
x_1 + &2x_2 =  & 5 
\end{Bmatrix}</script><script type="math/tex; mode=display">
\left\{\begin{matrix}
x+y-z = 4 \\
2x -3y+z=3 \\
-5x+2y-2z=1\\
\end{matrix}\right.</script><h3 id="消元步骤-（阶梯状）"><a href="#消元步骤-（阶梯状）" class="headerlink" title="消元步骤 （阶梯状）"></a>消元步骤 （阶梯状）</h3><p>$ \left{\begin{matrix}<br>x&amp;+y&amp;-z = 4 \<br>2x&amp; -3y&amp;+z=3 \<br>-5x&amp;+2y&amp;-2z=1\<br>\end{matrix}\right. $&amp;    —-2,3列消去X—-&gt; $ \left{\begin{matrix}<br>x&amp;+y&amp; -z = 4  \<br>&amp;5y&amp; -3z = 5 \<br>&amp;-7y&amp; +7z = -21\<br>\end{matrix}\right. $—-3列消去Y—-&gt; $  \left{\begin{matrix}<br>x&amp;+y&amp;-z = 4  \<br>&amp;5y&amp; - 3z = 5 \<br>&amp;&amp;z = -5\<br>\end{matrix}\right.  $</p>
<p>三平面交于一点</p>
<p><img src="assets/7.png" alt=""></p>
<h3 id="欠定方程组-（有多个解）"><a href="#欠定方程组-（有多个解）" class="headerlink" title="欠定方程组 （有多个解）"></a>欠定方程组 （有多个解）</h3><p> 三平面交于一条直线或者一个平面</p>
<p><img src="assets/8.png" alt=""></p>
<h3 id="不相容-无解"><a href="#不相容-无解" class="headerlink" title="不相容 无解"></a>不相容 无解</h3><p>三个平面不相交 或者 没有公共点，线，面</p>
<figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">线性代数和初等代数的区别:借助于矩阵,用计算机解决问题</span><br></pre></td></tr></table></figure>
      
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        <h2 id="食品配方问题"><a href="#食品配方问题" class="headerlink" title="食品配方问题"></a>食品配方问题</h2><p>甲乙丙丁四种原料营养比如下</p>
<div class="table-container">
<table>
<thead>
<tr>
<th></th>
<th style="text-align:center">甲</th>
<th style="text-align:center">乙</th>
<th style="text-align:center">丙</th>
<th style="text-align:center">丁</th>
</tr>
</thead>
<tbody>
<tr>
<td>蛋白质%</td>
<td style="text-align:center">20</td>
<td style="text-align:center">16</td>
<td style="text-align:center">10</td>
<td style="text-align:center">15</td>
</tr>
<tr>
<td>脂肪%</td>
<td style="text-align:center">3</td>
<td style="text-align:center">8</td>
<td style="text-align:center">2</td>
<td style="text-align:center">5</td>
</tr>
<tr>
<td>碳水化合物%</td>
<td style="text-align:center">10</td>
<td style="text-align:center">25</td>
<td style="text-align:center">20</td>
<td style="text-align:center">5</td>
</tr>
</tbody>
</table>
</div>
<p> 做出满足 蛋白质 15%， 脂肪 5%， 碳水化合物 12% 的食品， 求 上述四种原料配比 。</p>
<h2 id="解法"><a href="#解法" class="headerlink" title="解法"></a>解法</h2><h3 id="建立数学模型"><a href="#建立数学模型" class="headerlink" title="建立数学模型"></a>建立数学模型</h3><p>设四种原料占该食物的百分比分别为 $x_1$, $x_2$, $x_3$, $x_4$. 则:</p>
<script type="math/tex; mode=display">
\begin{Bmatrix}
 x_1+&  x_2+&  x_3+&  x_4=&  1  &  （总含量）&\\\\ 
 20x_1 +& 16x_2 + & 10x_3 + &  15x_4 =  & 15  & (蛋白质)&\\\\ 
 3x_1 + & 8x_2 + & 2x_3 + & 5x_4 =  &  5  & (脂肪)&\\\\ 
 10x_1 +&  25x_2 + &   20x_3 + & 20x_4 = &  12  & (碳水化合物)&
\end{Bmatrix}.</script><h2 id="线性方程组解的类型"><a href="#线性方程组解的类型" class="headerlink" title="线性方程组解的类型"></a>线性方程组解的类型</h2><pre class="mermaid">graph LR;
  A(线性方程组解的类型) --> B{是否有解} 
    B --> D(有唯一解)
    B --> E(有多个解)
    B --> F(无解)
    F --> G(找出近似解)

    D --> H{是否合理}
    E-->H
    H-->I(解集的性质)</pre>



<h2 id="线性方程组的类型"><a href="#线性方程组的类型" class="headerlink" title="线性方程组的类型"></a>线性方程组的类型</h2><pre class="mermaid">graph LR;
  A(线性方程组类型) --> B(适定方程组)
  B-->G(唯一解)
  A-->C(欠定方程组)
  C-->E(解不唯一)
  A-->D(超定方程组)
  D-->F(不存在精确解, 可求出近似解)</pre>
      
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  <ul class="article-tag-list"><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/数学模型/">数学模型</a></li><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/线性方程组解的类型/">线性方程组解的类型</a></li></ul>

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        <p>1.<br><figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br></pre></td><td class="code"><pre><span class="line">&gt;&gt; randintr(<span class="number">4</span>,<span class="number">5</span>)</span><br><span class="line"></span><br><span class="line"><span class="built_in">ans</span> =</span><br><span class="line"></span><br><span class="line">     <span class="number">6</span>     <span class="number">3</span>     <span class="number">9</span>     <span class="number">9</span>    <span class="number">-1</span></span><br><span class="line">     <span class="number">8</span>    <span class="number">-8</span>     <span class="number">9</span>     <span class="number">0</span>     <span class="number">8</span></span><br><span class="line">    <span class="number">-7</span>    <span class="number">-4</span>    <span class="number">-7</span>     <span class="number">6</span>     <span class="number">6</span></span><br><span class="line">     <span class="number">8</span>     <span class="number">1</span>     <span class="number">9</span>    <span class="number">-7</span>     <span class="number">9</span></span><br><span class="line"></span><br><span class="line">&gt;&gt; A=<span class="built_in">ans</span></span><br><span class="line"></span><br><span class="line">A =</span><br><span class="line"></span><br><span class="line">     <span class="number">6</span>     <span class="number">3</span>     <span class="number">9</span>     <span class="number">9</span>    <span class="number">-1</span></span><br><span class="line">     <span class="number">8</span>    <span class="number">-8</span>     <span class="number">9</span>     <span class="number">0</span>     <span class="number">8</span></span><br><span class="line">    <span class="number">-7</span>    <span class="number">-4</span>    <span class="number">-7</span>     <span class="number">6</span>     <span class="number">6</span></span><br><span class="line">     <span class="number">8</span>     <span class="number">1</span>     <span class="number">9</span>    <span class="number">-7</span>     <span class="number">9</span></span><br><span class="line"></span><br><span class="line">&gt;&gt; ref1(A)</span><br><span class="line"></span><br><span class="line"><span class="built_in">ans</span> =</span><br><span class="line"></span><br><span class="line">    <span class="number">8.0000</span>   <span class="number">-8.0000</span>    <span class="number">9.0000</span>         <span class="number">0</span>    <span class="number">8.0000</span></span><br><span class="line">         <span class="number">0</span>  <span class="number">-11.0000</span>    <span class="number">0.8750</span>    <span class="number">6.0000</span>   <span class="number">13.0000</span></span><br><span class="line">         <span class="number">0</span>         <span class="number">0</span>    <span class="number">2.9659</span>   <span class="number">13.9091</span>    <span class="number">3.6364</span></span><br><span class="line">         <span class="number">0</span>         <span class="number">0</span>         <span class="number">0</span>   <span class="number">-5.4483</span>   <span class="number">10.7586</span></span><br><span class="line"></span><br><span class="line">&gt;&gt; rref(A)</span><br><span class="line"></span><br><span class="line"><span class="built_in">ans</span> =</span><br><span class="line"></span><br><span class="line">    <span class="number">1.0000</span>         <span class="number">0</span>         <span class="number">0</span>         <span class="number">0</span>  <span class="number">-12.2222</span></span><br><span class="line">         <span class="number">0</span>    <span class="number">1.0000</span>         <span class="number">0</span>         <span class="number">0</span>   <span class="number">-1.4248</span></span><br><span class="line">         <span class="number">0</span>         <span class="number">0</span>    <span class="number">1.0000</span>         <span class="number">0</span>   <span class="number">10.4866</span></span><br><span class="line">         <span class="number">0</span>         <span class="number">0</span>         <span class="number">0</span>    <span class="number">1.0000</span>   <span class="number">-1.9747</span></span><br><span class="line"></span><br><span class="line">&gt;&gt; A=randintr(<span class="number">5</span>,<span class="number">6</span>,<span class="number">20</span>)</span><br><span class="line"></span><br><span class="line">A =</span><br><span class="line"></span><br><span class="line">     <span class="number">6</span>    <span class="number">11</span>     <span class="number">8</span>    <span class="number">13</span>    <span class="number">-3</span>     <span class="number">0</span></span><br><span class="line">   <span class="number">-19</span>    <span class="number">10</span>   <span class="number">-19</span>     <span class="number">8</span>    <span class="number">-5</span>    <span class="number">-2</span></span><br><span class="line">    <span class="number">14</span>    <span class="number">-4</span>    <span class="number">-9</span>    <span class="number">-7</span>    <span class="number">11</span>     <span class="number">6</span></span><br><span class="line">    <span class="number">18</span>     <span class="number">6</span>   <span class="number">-19</span>    <span class="number">18</span>    <span class="number">12</span>     <span class="number">9</span></span><br><span class="line">     <span class="number">7</span>   <span class="number">-13</span>   <span class="number">-17</span>   <span class="number">-19</span>   <span class="number">-13</span>    <span class="number">10</span></span><br><span class="line"></span><br><span class="line">&gt;&gt; rref(A)</span><br><span class="line"></span><br><span class="line"><span class="built_in">ans</span> =</span><br><span class="line"></span><br><span class="line">    <span class="number">1.0000</span>         <span class="number">0</span>         <span class="number">0</span>         <span class="number">0</span>         <span class="number">0</span>    <span class="number">0.3738</span></span><br><span class="line">         <span class="number">0</span>    <span class="number">1.0000</span>         <span class="number">0</span>         <span class="number">0</span>         <span class="number">0</span>   <span class="number">-0.0423</span></span><br><span class="line">         <span class="number">0</span>         <span class="number">0</span>    <span class="number">1.0000</span>         <span class="number">0</span>         <span class="number">0</span>   <span class="number">-0.2548</span></span><br><span class="line">         <span class="number">0</span>         <span class="number">0</span>         <span class="number">0</span>    <span class="number">1.0000</span>         <span class="number">0</span>   <span class="number">-0.0182</span></span><br><span class="line">         <span class="number">0</span>         <span class="number">0</span>         <span class="number">0</span>         <span class="number">0</span>    <span class="number">1.0000</span>   <span class="number">-0.1658</span></span><br></pre></td></tr></table></figure></p>
<ol>
<li><p>解方程组</p>
<figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br></pre></td><td class="code"><pre><span class="line">&gt;&gt; A = [ <span class="number">3</span> <span class="number">-2</span> <span class="number">-5</span> <span class="number">1</span>; <span class="number">2</span> <span class="number">-3</span> <span class="number">1</span> <span class="number">5</span>;<span class="number">1</span> <span class="number">2</span> <span class="number">0</span> <span class="number">-4</span>; <span class="number">1</span> <span class="number">-1</span> <span class="number">-4</span> <span class="number">9</span>]</span><br><span class="line"></span><br><span class="line">A =</span><br><span class="line"></span><br><span class="line">       <span class="number">3</span>             <span class="number">-2</span>             <span class="number">-5</span>              <span class="number">1</span></span><br><span class="line">       <span class="number">2</span>             <span class="number">-3</span>              <span class="number">1</span>              <span class="number">5</span></span><br><span class="line">       <span class="number">1</span>              <span class="number">2</span>              <span class="number">0</span>             <span class="number">-4</span></span><br><span class="line">       <span class="number">1</span>             <span class="number">-1</span>             <span class="number">-4</span>              <span class="number">9</span></span><br><span class="line"></span><br><span class="line">&gt;&gt; b = [<span class="number">3</span>; <span class="number">-3</span>; <span class="number">-3</span>; <span class="number">22</span>]</span><br><span class="line"></span><br><span class="line">b =</span><br><span class="line"></span><br><span class="line">       <span class="number">3</span></span><br><span class="line">      <span class="number">-3</span></span><br><span class="line">      <span class="number">-3</span></span><br><span class="line">      <span class="number">22</span></span><br><span class="line"></span><br><span class="line">&gt;&gt; [U,ip]=rref([A,b])</span><br><span class="line"></span><br><span class="line">U =</span><br><span class="line"></span><br><span class="line">       <span class="number">1</span>              <span class="number">0</span>              <span class="number">0</span>              <span class="number">0</span>             <span class="number">-1</span></span><br><span class="line">       <span class="number">0</span>              <span class="number">1</span>              <span class="number">0</span>              <span class="number">0</span>              <span class="number">3</span></span><br><span class="line">       <span class="number">0</span>              <span class="number">0</span>              <span class="number">1</span>              <span class="number">0</span>             <span class="number">-2</span></span><br><span class="line">       <span class="number">0</span>              <span class="number">0</span>              <span class="number">0</span>              <span class="number">1</span>              <span class="number">2</span></span><br><span class="line"></span><br><span class="line"></span><br><span class="line">ip =</span><br><span class="line"></span><br><span class="line">       <span class="number">1</span>              <span class="number">2</span>              <span class="number">3</span>              <span class="number">4</span></span><br></pre></td></tr></table></figure>
</li>
<li><p>交通流量</p>
<figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br></pre></td><td class="code"><pre><span class="line">&gt;&gt; A = [0 -1 0 0 1; 0 0 0 1 -1; 1 1 0 0 0;0 0 1 1 0; -1 0 1 0 0;]</span><br><span class="line"></span><br><span class="line">A =</span><br><span class="line"></span><br><span class="line">       0             -1              0              0              1</span><br><span class="line">       0              0              0              1             -1</span><br><span class="line">       1              1              0              0              0</span><br><span class="line">       0              0              1              1              0</span><br><span class="line">      -1              0              1              0              0</span><br><span class="line"></span><br><span class="line">&gt;&gt; b = [300; -100; 300; 700; 200]</span><br><span class="line"></span><br><span class="line">b =</span><br><span class="line"></span><br><span class="line">     300</span><br><span class="line">    -100</span><br><span class="line">     300</span><br><span class="line">     700</span><br><span class="line">     200</span><br><span class="line"></span><br><span class="line">&gt;&gt; [U,ip] = rref([A,b])</span><br><span class="line"></span><br><span class="line">U =</span><br><span class="line"></span><br><span class="line">       1              0              0              0              1            600</span><br><span class="line">       0              1              0              0             -1           -300</span><br><span class="line">       0              0              1              0              1            800</span><br><span class="line">       0              0              0              1             -1           -100</span><br><span class="line">       0              0              0              0              0              0</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">ip =</span><br><span class="line"></span><br><span class="line">       1              2              3              4</span><br></pre></td></tr></table></figure>
</li>
<li><p>解三次多项式</p>
</li>
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      <a class="article-title" href="/2019/05/17/English/College_life_in_the_internet_age/">College life in the internet age</a>
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        <p>The college campus, long a place of scholarship and frontiers of new technology, is being transformed into a new age of electronics by a fleet of laptops, smartphones and connectivity 24 hours a day</p>
<p>大学校园长期以来一直是学术和新技术前沿之地，如今正通过24小时不间断的笔记本电脑、智能手机和网络连接，将其转变为一个新的电子时代</p>
<ul>
<li><a href="https://fanyi.baidu.com/#en/zh/scholarship">scholarship</a>奖学金; 学问; 学术; 学术研究</li>
<li><a href="https://fanyi.baidu.com/#en/zh/frontiers">frontiers</a>国界; 边界; 边境; (尤指19世纪美国西部的)开发地区边缘地带，边远地区; 尖端，边缘; frontier的复数</li>
<li><a href="https://fanyi.baidu.com/#en/zh/new technology">new technology</a>新科技</li>
<li><a href="https://fanyi.baidu.com/#en/zh/transformed">transformed</a>使改变形态; 使改变外观; 使改观; transform的过去分词和过去式</li>
<li><a href="https://fanyi.baidu.com/#en/zh/fleet">fleet</a>舰队; 捕鱼船队; 全部军舰，海军; 跑得快的; 快速的</li>
<li><a href="https://fanyi.baidu.com/#en/zh/laptops">laptops</a>膝上型计算机; 便携式电脑; 笔记本电脑; laptop的复数</li>
<li><a href="https://fanyi.baidu.com/#en/zh/smartphones">smartphones</a>smartphone的复数</li>
<li><a href="https://fanyi.baidu.com/#en/zh/connectivity">connectivity</a>连接; 联结</li>
</ul>
<p>On a typical modern-day campus, where every buildings and most outdoor common areas offer wireless Internet access, one student takes her laptop everywhere. In class, she takes notes with it, sometimes instant-messaging or emailing friends if the professor is less than interesting. In her dorm, she instant-message her roommate sitting just a few feet away. She is tied to her smart-phone ,which she even uses to text a friend who lives one floor above her, and which supplies music for walks between classes</p>
<ul>
<li><a href="https://fanyi.baidu.com/#en/zh/modern-day">modern-day</a>现代的; 当代的; 现代版的，翻新的</li>
<li><a href="https://fanyi.baidu.com/#en/zh/with it">with it</a>时髦的，新派的; 警觉的；敏锐的；机敏的</li>
<li><a href="https://fanyi.baidu.com/#en/zh/less than">less than</a>毫不; 完全不; 一点都不</li>
<li><a href="https://fanyi.baidu.com/#en/zh/dorm">dorm</a>同 dormitory; 学生宿舍</li>
<li><a href="https://fanyi.baidu.com/#en/zh/roommate">roommate</a>室友; 同住一室的人</li>
<li><a href="https://fanyi.baidu.com/#en/zh/tied">tied</a>只租给雇工居住的; 系，拴，绑，捆，束; 将…系在…上; 束紧; 系牢; 捆绑; 打结，系扣; tie的过去分词和过去式</li>
<li><a href="https://fanyi.baidu.com/#en/zh/walks">walks</a>走; 行走; 步行; 徒步旅行; 散步; 陪伴…走; 护送…走; 散步的小路; 步行的路径; 步态; 步行速度; walk的第三人称单数和复数</li>
</ul>
<p>在一个典型的现代校园里，每座建筑和大多数户外公共区域都提供无线上网，一个学生带着她的笔记本电脑到处走。在课堂上，她用它做笔记，有时如果教授不那么有趣，她就用它发即时消息或电子邮件给朋友。在宿舍里，她给坐在几英尺外的室友发即时信息。她被绑在智能手机上，甚至可以用它给住在楼上的朋友发短信，还可以在课间播放音乐</p>
<p>Welcome to college life in the 21st century. where students on campus are electronically linked to each other , to professors and to their classwork 24/7 in an ever-flow river of information and communication. With many schools offering wireless Internet access anywhere on campus, colleges as a group have the most Internet-accessible spots in the world.</p>
<ul>
<li><a href="https://fanyi.baidu.com/#en/zh/college life">college life</a>大学生活</li>
<li><a href="https://fanyi.baidu.com/#en/zh/on campus">on campus</a>在校内</li>
<li><a href="https://fanyi.baidu.com/#en/zh/electronically">electronically</a>用电子方法; 用电子装置</li>
<li><a href="https://fanyi.baidu.com/#en/zh/each other">each other</a>互相，彼此</li>
</ul>
<p>Students say they really value their fingertip-access to the boundless amount of information online, and the ability to email to professors at 2 a.m and receive responses the next morning. “I always feel like I have a means of communication - in class and out of class”, say one engineering major.</p>
<ul>
<li><a href="https://fanyi.baidu.com/#en/zh/boundless">boundless</a>无限的; 无止境的</li>
<li><a href="https://fanyi.baidu.com/#en/zh/professors">professors</a>教授; 讲师，教员; professor的复数</li>
<li><a href="https://fanyi.baidu.com/#en/zh/a.m">a.m</a>上午; 文科硕士</li>
<li><a href="https://fanyi.baidu.com/#en/zh/responses">responses</a>回答，答复; 反应; 响应; 答唱咏，启应经; response的复数</li>
<li><a href="https://fanyi.baidu.com/#en/zh/morning">morning</a>早晨; 上午; 午夜至正午的时间; 在上午; 在早晨; 每天上午</li>
<li><a href="https://fanyi.baidu.com/#en/zh/feel like">feel like</a>摸起来像是……；有……的感觉; 想要……</li>
<li><a href="https://fanyi.baidu.com/#en/zh/means of communication">means of communication</a>通信工具</li>
<li><a href="https://fanyi.baidu.com/#en/zh/in class">in class</a>在上课; 在上课中</li>
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